cp's OEIS Frontend

A269650 Number of length-n 0..2 arrays with no adjacent pair x,x+1 repeated.

%I A269650 #12 Apr 21 2023 09:23:52
%S A269650 3,9,27,79,225,626,1710,4605,12259,32320,84504,219356,565816,1451349,
%T A269650 3704271,9412153,23818707,60055275,150913073,378064818,944442242,
%U A269650 2353140149,5848794543,14504575980,35894673012,88654500384,218560230944
%N A269650 Number of length-n 0..2 arrays with no adjacent pair x,x+1 repeated.
%H A269650 R. H. Hardin, <a href="/A269650/b269650.txt">Table of n, a(n) for n = 1..210</a>
%H A269650 Robert Israel, <a href="/A269650/a269650.pdf">Maple-assisted proof of recurrence</a>
%F A269650 Empirical: a(n) = 9*a(n-1) - 33*a(n-2) + 66*a(n-3) - 84*a(n-4) + 75*a(n-5) - 47*a(n-6) + 21*a(n-7) - 6*a(n-8) + a(n-9).
%F A269650 Empirical g.f.: x*(1 - 2*x + x^2 - x^3)*(3 - 12*x + 18*x^2 - 14*x^3 + 5*x^4 - x^5) / (1 - 3*x + 2*x^2 - x^3)^3. - _Colin Barker_, Jan 25 2019
%F A269650 Empirical recurrence verified (see link). - _Robert Israel_, Apr 19 2023
%e A269650 Some solutions for n=9:
%e A269650 ..1. .2. .2. .0. .0. .2. .1. .2. .2. .2. .0. .0. .2. .1. .1. .0
%e A269650 ..1. .1. .0. .0. .0. .0. .2. .1. .1. .2. .2. .2. .0. .1. .2. .0
%e A269650 ..1. .2. .2. .0. .0. .1. .0. .0. .0. .1. .2. .2. .1. .1. .2. .0
%e A269650 ..1. .1. .1. .0. .2. .0. .2. .1. .2. .2. .0. .2. .2. .1. .1. .1
%e A269650 ..0. .1. .0. .2. .1. .0. .2. .1. .2. .0. .2. .2. .1. .2. .0. .1
%e A269650 ..0. .0. .1. .0. .1. .2. .2. .2. .2. .2. .2. .0. .1. .2. .0. .1
%e A269650 ..2. .2. .2. .2. .2. .0. .2. .2. .1. .2. .2. .2. .0. .0. .2. .0
%e A269650 ..0. .2. .0. .0. .2. .0. .1. .2. .2. .0. .2. .0. .0. .0. .2. .2
%e A269650 ..2. .2. .0. .1. .0. .2. .1. .0. .2. .1. .2. .2. .0. .1. .2. .1
%p A269650 T:= Matrix(12,12):
%p A269650 for i from 1 to 12 do T[i,i]:= 1 od:
%p A269650 T[1,6]:= 1: T[3,8]:= 1:
%p A269650 T[5,11]:= 1: T[6,12]:= 1:
%p A269650 for i from 1 to 4 do T[i,i+8]:= 1; T[i+4,i]:= 1; T[i+8,i]:= 1; T[i+8,i+4]:= 1 od:
%p A269650 u:= <1,0,0,0,1,0,0,0,1,0,0,0>: v:= <1$12>:
%p A269650 seq(u^%T . T^i . v, i = 0 .. 50); # _Robert Israel_, Apr 19 2023
%Y A269650 Column 2 of A269656.
%K A269650 nonn
%O A269650 1,1
%A A269650 _R. H. Hardin_, Mar 02 2016